# generate PDF and CDF of GAMMA Distribution
shap = 3
scal = 1
x=seq(0,10,by=0.5)
# pdf of gamma distribution
pdf = dgamma(x, shape = shap, scale = scal)
# mean and median gamma distribution
mean.gamma = shap*scal
median.gamma = qgamma(0.5, shape = shap, scale = scal)
# plot pdf
plot(x,pdf,type = "l",main = "PDF of the GAMMA Distribution")
abline(v = mean.gamma, col = "red")
abline(v = median.gamma, col = "blue")
# plot cdf
x.gam.cdf = seq(0,15,by = 0.1)
cdf = pgamma(x.gam.cdf, shape = shap, scale = scal)
plot(x.gam.cdf,cdf,type = "l",main = "CDF of the GAMMA Distribution")
data = rgamma(100000,shape = shap, scale = scal)
hist(data,breaks = 100)
plot(ecdf(data))
log.gamma.data <- log(data)
hist(log.gamma.data,breaks = 100)
plot(ecdf(log.gamma.data))
art.mean = NA
geo.mean = NA
for (i in 1:1000) {
data = rgamma(n=100, shape = shap, scale = scal)
art.mean[i] = mean(data)
geo.mean[i] = exp(mean(log(data)))
}
plot(geo.mean,art.mean)
abline(0,1)
hist(geo.mean)
hist(art.mean)
data = rgamma(100000,shape = shap, scale = scal)
hist(data,breaks = 100)
plot(ecdf(data))
log.gamma.data <- log(data)
hist(log.gamma.data,breaks = 100)
plot(ecdf(log.gamma.data))
# generate PDF and CDF of lognorm Distribution
mea = -1
sd = 1
x.lognorm=seq(0,12,by=0.5)
# pdf of log normal distribution
pdf.lognorm = dlnorm(x.lognorm, meanlog = mea, sdlog = sd)
# mean and median log normal distribution
mean.lognorm = mean(dlnorm(x.lognorm, meanlog = mea, sdlog = sd))
median.lognorm = qlnorm(0.5, meanlog = mea, sdlog = sd)
# plot pdf
plot(x.lognorm,pdf.lognorm,type = "l",main = "PDF of the Log Normal Distribution")
abline(v = mean.lognorm, col = "red")
abline(v = median.lognorm, col = "blue")
# plot cdf
x.lognorm.cdf = seq(0,12,by = 0.1)
cdf.lognorm= plnorm(x.lognorm.cdf, meanlog = mea, sdlog = sd)
plot(x.lognorm.cdf,cdf.lognorm,type = "l",main = "CDF of the Log Normal Distribution")
data.lognorm = rlnorm(100000,meanlog = mea, sdlog = sd)
hist(data.lognorm ,breaks = 100)
plot(ecdf(data.lognorm ))
log.lognorm.data <- log(data.lognorm)
hist(log.lognorm.data,breaks = 100)
plot(ecdf(log.lognorm.data))
art.lognorm.mean = NA
geo.lognorm.mean = NA
for (i in 1:1000) {
data.lognorm = rlnorm(n=100, meanlog = mea, sdlog = sd)
art.lognorm.mean[i] = mean(data.lognorm)
geo.lognorm.mean[i] = exp(mean(log(data.lognorm)))
}
plot(geo.lognorm.mean,art.lognorm.mean)
abline(0,1)
hist(geo.lognorm.mean)
hist(art.lognorm.mean)
# generate PDF and CDF of Uniform Distribution
mi = 0
ma = 12
x.unif=seq(0,12,by=0.5)
# pdf of unifrom distribution
pdf.unif = dunif(x.unif,min = mi,max = ma)
# mean and median uniform distribution
mean.unif = (mi + ma)/2
median.unif = (mi + ma)/2
# plot pdf
plot(x.unif,pdf.unif,type = "l",main = "PDF of the Uniform Distribution")
abline(v = mean.unif, col = "red")
abline(v = median.unif, col = "blue")
# plot cdf
x.unif.cdf = seq(0,12,by = 0.1)
cdf.unif = punif(x.unif.cdf, min = mi,max = ma)
plot(x.unif.cdf,cdf.unif,type = "l",main = "CDF of the Uniform Distribution")
data.unif = runif(100000, min = mi,max = ma)
hist(data.unif ,breaks = 100)
plot(ecdf(data.unif ))
log.unif.data <- log(data.unif)
hist(log.unif.data,breaks = 100)
plot(ecdf(log.unif.data))
unif.art.mean = NA
unif.geo.mean = NA
for (i in 1:1000) {
unif.data = runif(n=100, min = mi,max = ma)
unif.art.mean[i] = mean(unif.data)
unif.geo.mean[i] = exp(mean(log(unif.data)))
}
plot(unif.geo.mean,unif.art.mean)
abline(0,1)
hist(unif.geo.mean)
hist(unif.art.mean)
Caption: Image of GAMA.
AM >= GM
Caption: Image of log.
log(E[X]) >= E[log(x)]